A152829 Numbers k whose squares can be written in exactly one way as a sum of three squares: k^2 = a^2 + b^2 + c^2 with 1 <= a <= b <= c.

3, 6, 7, 12, 13, 14, 24, 26, 28, 48, 52, 56, 96, 104, 112, 192, 208, 224, 384, 416, 448, 768, 832, 896, 1536, 1664, 1792, 3072, 3328, 3584, 6144, 6656, 7168, 12288, 13312, 14336, 24576, 26624, 28672, 49152, 53248, 57344, 98304, 106496, 114688, 196608, 212992, 229376

Offset

1,1

Comments

Numbers k such that k^2 is in A025321. - Joerg Arndt, Mar 22 2022

2k is a term iff k is also a term, so the conjecture from Colin Barker (see Formula) is true iff 3, 7, and 13 are the only odd terms. - Jon E. Schoenfield, Mar 22 2022

Links

Table of n, a(n) for n=1..48.

M. A. Fajjal, Posting in sci.math.symbolic

Formula

Guessed o.g.f.: x*(x^4 + 6*x^3 + 7*x^2 + 6*x + 3)/(1 - 2*x^3).

{k: A025427(k^2)=1}. - R. J. Mathar, Dec 15 2008

Conjecture: a(n) = 2*a(n-3) for n > 5. - Colin Barker, Mar 12 2012

Example

9 is not in this sequence because 9^2 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
7 is in this sequence because 7^2 = 2^2 + 3^2 + 6^2 is the only way to write 7^2 as a sum of three squares.

Prog

(C)
#include <stdio.h>
#include <stdlib.h>
int main (int argc, char *argv[]) {
    long n, n2, a, a2, b, b2, c, c2; int s = 0; n=atol(argv[1]); n2=n*n;
    for (a=1; a<n; a++) {
        a2 = a*a;
        for (b=a; (b2=b*b)<n2-a2; b++)
            for (c=b; (c2=c*c)<=n2-a2-b2; c++) if (a2+b2+c2==n2) s++;
    }
    return (s==1?0:1);
}
# Then the bash-line
for (( i=1; i<1000; i++ )); do ./a.out $i && echo -n $i, ; done > 3sq.txt
# gives the terms less than 1000

Crossrefs

Cf. A025321.

Sequence in context: A186083 A093906 A332812 * A325430 A104463 A072757

Adjacent sequences: A152826 A152827 A152828 * A152830 A152831 A152832

Keyword

nonn

Author

Peter Pein (petsie(AT)dordos.net), Dec 13 2008

Extensions

a(25)-a(36) (from comment) verified and added by Donovan Johnson, Nov 08 2013

a(37)-a(48) from Jon E. Schoenfield, Mar 22 2022

Status

approved